Rare-Event Study for Stochastic Partial Differential Equations
DescriptionRare events are those outcomes in a stochastic system that rarely happen but once they happen, they may have drastic consequences, such as material failure or pollution breakout. They are closely related to the stability or risk measure of the system under extreme conditions. Stochastic partial differential equations (SPDE) are partial differential equations (PDE) that contain random forcing or coefficients and they are extensively used in various fields of science and engineering to incorporate random effects in physical systems. In this proposal, we focus on a special type of SPDE — elliptic PDEs with random coefficients. There is a considerable amount of literature about SPDE in mathematics analysis and numerical techniques. However, traditional stochastic theories and algorithms mainly concern the averaged effect of the randomness. Rare events occur only at certain locations in the random space which are far away from where the typical behaviors occur.This project aims at the challenges arising from the analysis and computation of the rare events for elliptic partial differential equations with random coefficients which are of great interests in applications, such as material failure, spreading of pollutants in ground water, etc. The random coefficients are modeled as random fields related to Gaussian random field. The solution of the equation is a complicated functional of the input Gaussian field. The rare events of interests correspond to the behaviors of the solution which significantly deviate from the typical states. We will discuss three scenarios of rare events. Different ideas and techniques will be developed for these different scenarios. The common objective is to derive asymptotic results and to design the importance sampling Monte Carlo method to calculate the small probabilities of the rare events. The theory of large deviation principle and Gaussian random function as well as some analysis techniques for elliptic partial differential equation and heuristic arguments will be exploited and unified for our purpose. The numerical method will combine the PDE-related numerical techniques with the rare event simulation schemes, such as the importance sampling method guided by the asymptotic analysis and other types of heuristics.
|Effective start/end date||1/01/14 → 12/12/17|